Questions tagged [dual-spaces]
The dual space of a vector space $V$ over a field $k$ is the vector space of all linear maps from $V$ into $k$.
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Is the duality product of dual spaces unique?
Let $\Omega \subset \mathbb{R}^n$. Consider, for example, the Sobolev space $H^1_0(\Omega)$. It is known that the dual is $H^{-1}(\Omega)$ and is Banach with respect to the operator norm
$$||f||_{-1} =...
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Defining the Fourier projection operator on $H^{-1}$ so that it moves freely on the dual pairing?
Let us consider the circle $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ and for any $f \in L^2(\mathbb{T},\mathbb{C})$, define $P_N : L^2(\mathbb{T},\mathbb{C}) \to L^2(\mathbb{T},\mathbb{C})$ as
\begin{...
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How can a non-reflexive space have a reflexive subspace?
I know that this can happen (take any one-dimensional subspace, for instance), but I had the following thought while reading Kadison and Ringrose (Fundamentals of the Theory of Operator Algebras, Vol ...
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On the quadratic coalgebras
It is well known that a quadratic algebra $A(V,R)$ is the quotient of the free associative algebra $T(V)$ over a vector space $V$ by the two-sided ideal $(R)$ generated by $R\subseteq V^{\otimes 2}$, ...
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Prove that dual of a unital Banach algebra is nonempty using resolvent and Liouville's theorem
Let $\mathcal{A}$ be a unital Banach algebra over complex numbers $\mathbb{C}$. For every $a \in \mathcal{A}$, let $\sigma(a)$ be the spectrum of $a$. Define the resolvent of $a$ to be $ R(a,z) = (...
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The correct way to map a column vector into a row vector. [closed]
I'm a physics student studying linear algebra and the matrix representation of usual and dual vectors. I have several questions related to this topic, and I'm not sure if they make sense.
Firstly, the ...
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Image and kernel of transposed maps
Let $U,V,W$ be vector spaces over a field $K$ and $\varphi: U\to V,$ $\psi: V\to W$ linear maps. Given that
$\text{ker}(\psi)=\text{im}(\varphi)$ show that $\ker(\varphi^{tr})=\operatorname{im}(\psi^{...
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On the definition of pullback in linear algebra
Here is the definition of pullback maps in linear algebra from Mclnerney's First Steps in Differential Geometry:
Let $\Psi: V\to W$ be a linear transformation and let $\Psi^*:W^*\to V^*$ be given by $(...
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Dual space of a topological vector space that doesn't separate points?
While I am studying FUNCTIONAL ANALYSIS by Walter Rudin, I found the following corollary.
Now, I wonder how the dual space(the set of all continuous linear functionals on $X$) of some pathetic ...
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Definitions of non degenerate bilinear forms
Referring to this Wikipedia page, The definition of a non degenerate bilinear form
is given as a bilinear form $f(x,y)$ such that $v \to (x \to f(x,v))$ is an isomorphism.
We are also told that the ...
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Identifying dual space of a hyperplane in Euclidean spaces
I am considering a hyperplane in $\mathbb R^n$ given by
\begin{equation}
V=\left\{ (v_1,\ldots,v_n)\in\mathbb R^n: \sum_{i=1}^n v_i =0 \right\}.
\end{equation}
Question: Is there an explicit ...
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How to show duality of $l_1$ and $l_\infty$ balls using linear programming duality.
Let $\| x \|_1 = \sum_{i=1}^n |x_i|$ and $\| x \|_\infty = \max_{i=1}^n |x_i|$ be the $l_1$ and $l_\infty$ norms on $\mathbb{R}^n$.
For any given norm on $\mathbb{R}^n$, the associated dual norm (in ...
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What is the relationship between convex conjugate and polar set?
Given a duality $\left<X^*,X\right>$ over field $\mathbb R$, and any set $A \subseteq X$, the polar set of $A$ is defined as \begin{align}A^\circ = \{x^* \in X^* | \left<x^*,x\right> \leq ...
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Why is the transpose so useful?
I am learning linear algebra using the textbook Linear Algebra Done Right, trying to understand the subject through a logical, pure math perspective. I'm, simultaneously, learning applied linear ...
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How do we know $T'(\psi_j)=\psi_j\circ T$, where $T:V\to W$, $T': W'\to W'$ is dual map of $T$, and $\psi_j\in W'$?
Sometimes I have trouble with the notation in the book "Linear Algebra Done Right" by Axler.
Currently, my issue is with the following theorem (Theorem 3.114 in the 3rd edition book)
Let $...
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What does the transpose of a linear transformation represent? [duplicate]
I'm struggling to understand what the transpose of a linear transformation represents. My textbook's motivation for this wasn't very helpful. All it did was ask, "Is there a linear transformation ...
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Let $M$ be a smooth manifold. Does there exist a canonical isomorphism between $\Gamma(TM)^*$ and $\Gamma(T^*M)$?
Let $M$ be a smooth manifold and for each point $p \in M$, let $T_pM$ denote the tangent space at $p \in M$. We define the set $TM = \bigsqcup_{p \in M}T_pM$ and equip it with the initial topology and ...
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Dimension of range($\Gamma$) where $\Gamma$ is the linear map from $V'\to\mathbb{F}$ defined $\Gamma(\varphi)=(\varphi(v_1),...,\varphi(v_m))$?
Suppose $V$ is finite-dimensional and $v_1,...,v_m\in V$. Define a linear map $\Gamma: V'\to \mathbb{F}^m$ by
$$\Gamma(\varphi)=(\varphi(v_1),...,\varphi(v_m))\tag{1}$$
where $V'$ is the dual map of $...
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Is the dual of a unitary operator on a Hilbert space always a continous linear operator?
I have encountered this in a course (not functional analysis), but I am wondering whether this is a general fact.
Thank you very much in advance for any input!
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dual norm optimization problem, how to dervie the second formula from the first one?
There is a post that someone answered it already, but I think his/her answer is how to validate the formula, not the way to derive the formula.
A dual norm optimization problem
This formula comes from ...
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Reproducing kernel Hilbert space induced by $k(x, y) = \delta_{x, y}$, where $\delta$ is the Kronecker delta
I am trying to find the reproducing kernel Hilbert space induced by the symmetric positive definite (and bounded and measurable) kernel
$$
k \colon X \times X \to \{ 0, 1 \}, \qquad
(x, y) \mapsto %\...
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Why is this a good way to think geometrically about linear functionals?
In my textbook (Szekeres's A Course in Modern Mathematical Physics) the author writes
Perhaps the best way to visualize a linear functional is as a set of parallel planes of vectors determined by $\...
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What is the dual of integration over the unit interval?
Let $C[0,1]$ be the collection of continuous functions on the unit closed interval $[0,1]$.
It is a vector space with respect to the usual addition and scaler multiplication.
Let $\Phi:C[0,1]\to\...
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If all compositions of a vector-valued function with its linear functionals are analytic, is the function itself analytic?
This is taken from Conway's A course in functional Analysis (p. 198, Exercise 4):
Let $\mathscr{X}$ be a Banach space and $G \subset \mathbb{C}$ open. If $f: G \rightarrow \mathscr{X}$ is such that ...
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Quotient spaces and annihilators of inifinite dimensional vector spaces
Let $V$ be a vector space and $W$ be the dual space of $V$. For a subspace $U$ of $V$, we define
$$
U^0 = \{f\in W \mid f(u)=0 \text{ for all } u\in U \}.
$$
We are looking for an example in which
$$
...
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Dual of the annihilator isomorphic to quotient space
There are many resources online detailing the isomorphism between the annihilator $U^0:=\{f\in V':f(U)=0\}$ and the dual of the quotient space, $(V/U)':=Hom(V/U,\mathbb{R})$.
However, my lecture notes ...
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How to understand the dual of determinants
We consider the determinant of a $n\times n$ square matrix as a linear transformation.
$$\det : M_{n\times n}\to \mathbb{C}.$$
How to discribe its dual $\det ^*$?
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Weak continuity for a possibly nonlinear functional on a normed space
I am reading the fifth chapter ("Dual Spaces") from David Luenberger's Optimization by Vector Space Methods (1969). In Section 5.10, weak continuity for possibly nonlinear functionals on ...
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Uniqueness of Fourier transform of a measure [duplicate]
Let $C_0(\mathbb{R}^d)$ be the space of real-valued continuous functions on $\mathbb{R}^d$ that vanish at infinity. By Riesz–Markov–Kakutani representation theorem, the dual of $C_0(\mathbb{R}^d)$ is $...
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Definition of weak* continuity for possibly nonlinear functionals on normed dual spaces
I am reading the fifth chapter (on Dual Spaces) from David Luenberger's Optimization by Vector Space Methods. In Section 5.10, the author has defined weak continuity for possibly nonlinear functionals ...
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Let $f:V \times W \to \mathbb F$ and $s:V \to W^*, r:W \to V^*$. Find $[s]^B_{C^*}, [r]^C_{B^*}$.
Let $f:V \times W \to \mathbb F$ bilinear map and $s:V \to W^*, r:W \to V^*$ linear transformations s.t: $s(v)(w) = f(v,w)$ and $r(w)(v) = f(v,w), \forall v \in V$ and $\forall w \in W$. Let $B$ an ...
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What is dual space, in simple words?
I noticed that the rank of a tensor is a kind of "two-dimensional" property - the covariant components come first, then the contravariant. If I understand correctly, these components refer ...
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Confusion on proof of existence of ismophism between bilinear form on $L$ and dual space $\mathcal{L}(L,L^*)$
Theorem: There exists an isomorphism between the space of bilinear forms $\varphi$ on vector space $L$ and space $\mathcal{L}(L,L^*)$ of linear transformations $\mathcal{A}:L\mapsto L^*$.
I'm ...
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Prove that the bilinear function $(l,x)$ gives a natural identification of $X$ with $X’’$.
I’m attempting a proof of Theorem 3 below. I would like to know if my proof is valid. If it is, is there a “cleaner” proof? If it is not valid, can you push me in the right direction? Please first ...
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Dual space of Sobolev spaces induced by Elliptic equations
Given a domain $\Omega \subset \mathbb{R}^n$, it is well-known that the dual space of the Sobolev space $W_0^{1,2}(\Omega)$ (also denoted by $H^1_0(\Omega)$ by many people) is, by definition, the ...
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Relationship between the dual of a subspace and its annihilator
Let $V$ be a finite-dimensional vector space and let $W$ be a subspace of $V$. I shall use the following notation: $V^*$ will be the dual space of $V$, $Wº$ will be the annihilator of $W$, that is the ...
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Proving that a map is weak* continuous
Let $X$ be a compact space, $C(X)$ the space of continuous functions on $X$, and $\mathscr{M} = C(X)^*$, the dual of $C(X)$ which we identify with the space of all complex Baire measures. Let $$\...
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Proof: If T (T') is surjective then T' (T) is injective.
I am a little unsure about my proof. I wanted to ask if this proof is correct.I would appreciate any suggestions and improvements. Futhermore I dont yet fully understand why in (a) from the ...
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Is there any bounded linear map $T:X\to X$ such that $T$ is injective, $T^{\ast}$ is not surjective and $R(T^{\ast})$ is closed?
Here $X$ is Banach space and $T^{\ast}$ is the continuous dual(adjoint) of $T,R(T^{\ast})$ stands for the range of $T^{\ast}.$
The following is what I get:
If such a $T$ exists, then $X$ cannot be ...
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Are the weakly bounded subsets of the double dual bounded for the natural topology?
I'm trying to get a better understanding of the topologies on the double dual of a Hausdorff locally convex space (l.s.c.). The following question has then come up, and I'm unable to find an answer.
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Geometric Hahn-Banach theorem and linear operators [duplicate]
I've got this problem. $(X, ||\cdot||)$ is a normed vectorial space. Then, $l_0, l_1, ..., l_n$ are linear operators in $X^*$ (dual space) and for each $i\in \{0,1,...,n\}$, let $ker\:l_i = \{x\in X | ...
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Proving that a function $G: X\rightarrow \mathbb{R}^{n+1}$ is continuous.
$(X, ||\cdot||)$ is a normed vector space, and $l_0, l_1, ..., l_n$ are linear operators in $X^*$. And $G$ is defined like this. $G:X\rightarrow R^{n+1}: x\rightarrow (l_0(x), l_1(x), ..., l_n(x))$. ...
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How do I prove that a function is well defined? in my case, a continuous linear function.
This is my problem: Let $X$ and $Y$ be two normed vectorial spaces and let $L: X \rightarrow Y$ be a linear continuous functional. Let's now define $L^*: Y^* \rightarrow X^*$ like the the function ...
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Dual norm of integral vector norm
For $x\in\mathbb{C}^n$, consider the following norm:
$\|x\|_{B} := \int_{B}|\langle x,y\rangle| \ d\mu(y),$
where $B:=\{y\in\mathbb{C}^n \ | \ \|y\|_2 \leq 1\}$ is the euclidean-norm ($\|\circ\|_2$) ...
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Necessary and sufficient condition to write a quadratic form on a finite-dimensional real vector space as a product of two linear functionals
I have come across a tricky linear algebra problem. We want to prove that a quadratic form $q$ on a finite dimensional real vector space $V$ can be expressed as $q(v) = f_1(v)f_2(v) \iff r + |\sigma| \...
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Show that multiplicative functional are extremal points in a $C^*$-algebra.
I found the following exercise in my lecture notes: Let $\mathcal{A}$ be a $C^*$-algebra. Then every multiplicative functional is an extremal point of $K_1^{\mathcal{A}'}(0)$.
Other than writing $m = ...
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Weak*-separability of the unit ball of $X’$ and density characters and cardinalities of $X$ and $X’$
Let $X$ be a Banach space, $X’$ be its continuous dual such that its unit ball is weak*-separable. I’ve been wondering what can be said about the density characters (which I’ll denote by $d$) and ...
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How can a vector dual space be used to model division?
A couple of months ago I posted a question about how to formally define systems of units, and someone posted this fascinating response, explaining that each base unit can be the basis of a single ...
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If $L_n \rightarrow L \in X^*$ in the weak* sense and $x_n \rightarrow x$ in norm, does $L_n(x_n) \rightarrow L(x)$?
Let $X$ be a Banach space and $X^*$ its dual. Let $\{L_n\}$ be a sequence in $X^*$ such that $L_n \rightarrow L$ in the weak* sense, and suppose $\{x_n\}$ is a sequence in $X$ such that $x_n \...
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Containment of kernels of continuous seminorms
Let $X$ be a locally convex space, whose topology is defined by a family of seminorms $\mathcal{P}$.
Fact. For every continuous seminorm $q: X\to \mathbb{F}$ and positive number $\epsilon>0$, the ...
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